Lemma 2.13 (Berge's lemma for paths [1]}) 

Let \(G\)  be a graph and \(M\)  be a matching of \(G\) . If \(P\)  is an augmenting
path for \(M\)  – i.e. an alternating path whose endpoints are unmatched –
then \(M \triangle P\)  is a matching and \(|M \triangle P| = |M| + 1\) .
Conversely, if \(M\)  is a matching with \(|M^{\prime }| > |M|\) , then \(M \triangle M^{\prime }\)  is a
vertex-disjoint union of:
